Question

Remove parentheses and simplify. $$a^{2} b(2 a+b-a b)$$

Short answer

The simplified expression is \(2a^{3}b + a^{2}b^{2} - a^{3}b^{2}\).

Step-by-step solution

Distribute the Monomial

To remove the parentheses, distribute the monomial term outside the parentheses, which is \(a^{2}b\), across each term inside the parentheses. This means you will multiply \(a^{2}b\) by \(2a\), \(b\), and \(-ab\) separately.

Multiply Terms

Multiply \(a^{2}b\) by \(2a\) to get \(2a^{3}b\). Multiply \(a^{2}b\) by \(b\) to get \(a^{2}b^{2}\). Multiply \(a^{2}b\) by \(-ab\) to get \(-a^{3}b^{2}\).

Combine Like Terms

After distribution, we have \(2a^{3}b + a^{2}b^{2} - a^{3}b^{2}\). There are no like terms to combine, so this is the simplified expression.

Key concepts

Distributing Monomials

Understanding how to effectively distribute monomials across other terms within parentheses is a fundamental skill in algebra. Consider the expression \(a^{2}b(2a+b-ab)\). The first step is to apply the distributive property, which states that a term multiplied by a sum or difference (terms inside the parentheses) is equal to the sum or difference of the products. When distributing the monomial \(a^{2}b\), you perform individual multiplications with each term inside the parentheses.

To simplify \(a^{2}b(2a+b-ab)\), distribute \(a^{2}b\) to each of the three terms within the parentheses:

• \(a^{2}b \times 2a = 2a^{3}b\)
• \(a^{2}b \times b = a^{2}b^{2}\)
• \(a^{2}b \times (-ab) = -a^{3}b^{2}\).

By doing this systematically, you ensure that no term is left behind and the expression is expanded correctly.

Multiplying Terms

Multiplying terms, particularly when working with monomials and polynomials, involves applying the laws of exponents. In the previous example, the term \(a^{2}b\) is multiplied by each term within the parentheses. This requires you to multiply the coefficients and add the exponents of like bases.

Here’s a breakdown of how the multiplication is performed:

• For \(a^{2}b \times 2a\), multiply the coefficients (\(1 \times 2\)) and add the exponents of 'a' (\(2 + 1\)) to get \(2a^{3}b\).
• For \(a^{2}b \times b\), there is no 'a' in the second term, so you simply multiply coefficients and write the bases with their respective exponents to get \(a^{2}b^{2}\).
• For \(a^{2}b \times (-ab)\), multiply the coefficients (\(1 \times -1\)) and add the exponents of 'a' (\(2 + 1\)) and 'b' (\(1 + 1\)) to get \(a^{3}b^{2}\).

Combining Like Terms

After multiplying terms, the next step is to combine like terms to simplify the expression. Like terms have identical variables raised to the same power. In the distributed expression \(2a^{3}b + a^{2}b^{2} - a^{3}b^{2}\), we look for terms that can be combined.

A quick review of the terms reveals that there are no like terms present. As such:

• \(2a^{3}b\) is the only term with the variables \(a^{3}\) and \(b\).
• \(a^{2}b^{2}\) is distinct because it has \(a^{2}\) and \(b^{2}\).
• \(-a^{3}b^{2}\) is unique with \(a^{3}\) and \(b^{2}\), and it is subtracted.

This tells us that no further simplification is possible by combining terms, and the expression \(2a^{3}b + a^{2}b^{2} - a^{3}b^{2}\) is already in its simplest form.